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I am currently studying Newton-Raphson Method. I feel that I understand the concept of it. Somehow, I am facing some question in my head about how to actually apply it.

The questions that I have are below - How should I decide the first initial value? - How should I find all the roots on x-axis?, how should I set the ranges to find them separately?

Please, let me here your expertise. I am sorry if I have tagged my question in the wrong places. Thank you.

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  • $\begingroup$ To find the other roots, after your first approximation, you can divide the original equation by the factor you found, of course for algebraic equations only. $\endgroup$ – tpb261 Jul 4 '14 at 11:01
  • $\begingroup$ Can you please take an example for me? $\endgroup$ – user122358 Jul 4 '14 at 11:33
  • $\begingroup$ Let's say the equation is $x^3+3x^2+3x+1=0$ :D. One root is found to be -1. Then divide the original expression by $x+1$ to get $x^2+2x+1=0$. By observation, you can see that x=-1 is a triple root, but the program can't so, as a general rule, we have to divide the original expression by the factor. $\endgroup$ – tpb261 Jul 4 '14 at 11:55
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It will depend on the application. In most practical problems, you are likely to have some idea of the order of magnitude of the solution you expect to find. You take the initial value to be the best guess you have available. If you're lucky, Newton-Raphson might still work even if this initial guess is quite far from the actual solution. If you're unlucky, you can try another guess.

It sometimes helps if you can isolate the roots in intervals. If you can find $a < b$ such that one of $f(a)$ and $f(b)$ is positive and the other negative (and your function is continuous), you know that there is a solution somewhere in the interval $(a,b)$. If in addition $f$ is monotone on this interval, you know that there is only one solution there.

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  • $\begingroup$ Preferably (to me, at least !), start at a point $x_0$ such that $f(x_0) f''(x_0)>0$ $\endgroup$ – Claude Leibovici Jul 4 '14 at 7:57
  • $\begingroup$ ... hopefully one where this condition holds all the way to a root. Then Newton-Raphson will converge monotonically to that root. $\endgroup$ – Robert Israel Jul 4 '14 at 15:25
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Actually guessing proper initial value is helpful to find the root in less iteration. Improper guess of initial value can increase the number of iteration but surely generate the root(if the root can be found by Newton Raphson Method) after some more iteration.

I found an easy way to guess comparatively more suitable initial value: Draw the graph of the function and consider a point close to the zero crossing point of the function in X axis.

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